Question: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $a \neq 0$. $y = \dfrac{a}{9(2a + 9)} \div \dfrac{4a}{8(2a + 9)} $
Explanation: Dividing by an expression is the same as multiplying by its inverse. $y = \dfrac{a}{9(2a + 9)} \times \dfrac{8(2a + 9)}{4a} $ When multiplying fractions, we multiply the numerators and the denominators. $y = \dfrac{ a \times 8(2a + 9) } { 9(2a + 9) \times 4a } $ $ y = \dfrac{8a(2a + 9)}{36a(2a + 9)} $ We can cancel the $2a + 9$ so long as $2a + 9 \neq 0$ Therefore $a \neq -\dfrac{9}{2}$ $y = \dfrac{8a \cancel{(2a + 9})}{36a \cancel{(2a + 9)}} = \dfrac{8a}{36a} = \dfrac{2}{9} $